An electric power steering control apparatus in related art includes steering torque detecting means for detecting a steering torque by an operator, a torque controller that computes an auxiliary torque current for assisting the steering torque based on the detected steering torque signal, a motor that generates a torque for assisting the steering torque, vibration velocity estimating means for estimating a motor vibration velocity in a rotational direction of the motor, a damping controller that computes a damping current to be added to the auxiliary torque current using an estimation value of the estimated motor vibration velocity, and motor current detecting means. The vibration velocity detecting means in the rotational direction is current steering component removing means for removing a component by steering from a detection value or a command value of a current passed through the motor, torque steering component removing means for removing a component by steering from an output of the steering torque detecting means, and a rotational speed observer that is adapted for a vibration equation having moment of inertia of the motor as an inertial term and rigidity of a torque sensor as a spring term and computes the estimation value of the vibration velocity in the motor rotational direction based on the motor current output from the current steering component removing means and the steering torque output from the torque steering component removing means (for example, see Patent Document 1.    Patent Document 1: JP-A-2000-168600 (page 17, Amendment 1, claim 8)